In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent $λ_H$ satisfying $λ_H(θ)=\sum_{κ\in H\setminus\{0\}}\bigl(1-\Reθ(κ)\bigr)ν(κ),$ for a symmetric $ν\in\ell^1(H\setminus\{0\})$. This gives a symmetric jump process on $H$. The exponent $λ_H$ determines heat kernels, which define reproducing kernel Hilbert spaces and their associated semimetrics. Convex orders on the mixing measures give monotonicity for the kernels, Hilbert spaces, and semimetrics.